KöMaL Problems in Physics, December 2025
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Problems with sign 'M'Deadline expired on January 15, 2026. |
M. 445. Measure the space-filling properties of a given granular material (e.g. rice, barley, etc.). To what extent does this depend on the preparation of the system (e.g., compaction, shaking, etc.)?
(6 pont)
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Problems with sign 'G'Deadline expired on January 15, 2026. |
G. 905. A row boat is moving downstream at a speed of 8 m/s relative to the shore. A motorboat is coming upstream towards the row boat at a speed of 10 m/s relative to the water. Ten seconds after they meet, they are 160 meters apart. What is the speed of the river?
(3 pont)
solution (in Hungarian), statistics
G. 906. The density of salt water is \(\displaystyle 1025~\mathrm{kg}/\mathrm{m}^3\), while that of fresh water is \(\displaystyle 1000~\mathrm{kg}/\mathrm{m}^3\).
a) How does the buoyant force exerted on a ship change when the ship leaves the river mouth and sails out into the open sea?
b) Calculate how many kilograms of salt are contained in \(\displaystyle 1~\mathrm{m}^3\) of seawater, given that the density of sea salt is \(\displaystyle 2200~\mathrm{kg}/\mathrm{m}^3\). Assume that mixing salt and water does not cause any change in volume.
(3 pont)
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G. 907. Vertex \(\displaystyle A\) of the \(\displaystyle ABC\) equilateral triangle-shaped plate, which has uniform mass distribution and mass \(\displaystyle m=0.7~\mathrm{kg}\), is connected to the vertical wall with a hinge, as shown in the figure. Vertex \(\displaystyle B\), which is the vertex at the other end of the horizontal side \(\displaystyle AB\) of the triangle, is connected to the wall by a string. The string makes an angle of \(\displaystyle \varphi=60^\circ\) with the horizontal.
a) What is the tension in the string?
b) What is the magnitude and the direction of the force exerted on the hinge by the triangle-shaped plate?
(4 pont)
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G. 908. How much heat must be transferred to a copper ball with a diameter of \(\displaystyle 4.00~\mathrm{cm}\) and a temperature of \(\displaystyle 15~^\circ\mathrm{C}\) so that it cannot pass through a hole with a diameter of \(\displaystyle 4.02~\mathrm{cm}\)?
(3 pont)
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Problems with sign 'P'Deadline expired on January 15, 2026. |
P. 5688. From a given point, a heavy body is thrown at an angle \(\displaystyle \alpha\) with an initial velocity \(\displaystyle v_0\). Give the equation of the trajectory in the oblique coordinate system, where one axis points in the direction of the initial velocity and the other in the direction of the gravitational force. Can the constant in the equation be related to the ``parameter'' \(\displaystyle p\) of the parabolic trajectory?
(4 pont)
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P. 5689. Consider the Earth's orbit around the Sun as a circle with a radius of 150 million kilometres. Suppose that a comet in the plane of the ecliptic approaches the Sun along a parabolic path such that it intersects the Earth's orbit twice, and the length of the straight line connecting the two points of intersection is exactly the diameter of the Earth's orbit. (Fortunately, the Earth misses the comet by a wide margin when it moves close to the Earth's orbit.)
a) How close does the comet get to the centre of the Sun, and what is its speed at that position?
b) What is the angle between the Earth's orbit and the comet's orbit?
(5 pont)
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P. 5690. A small body of weight \(\displaystyle G\) rests on a slope of angle \(\displaystyle \alpha\). The coefficient of friction between the body and the slope is \(\displaystyle \mu>\tan\alpha\).
What is the minimum force needed to move the body, and in what direction should it be exerted?
(5 pont)
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P. 5691. Determine the moment of inertia of a thin, equilateral triangle-shaped plate of uniform mass distribution, with mass \(\displaystyle m\), and side length \(\displaystyle a\), with respect to an axis, which goes through one of the vertices of the triangle, and
a) which is perpendicular to the plane of the plate,
b) which coincides with the altitude of the triangle,
c) the axis is perpendicular to the previously described two axes.
(5 pont)
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P. 5692. A given quantity of monatomic ideal gas is quasi-statically taken from its initial state of pressure \(\displaystyle p_0\) and volume \(\displaystyle V_0\) to its final state of pressure \(\displaystyle p_0\) and volume \(\displaystyle 2V_0\). The process is chosen so that the temperature of the gas never decreases and the gas never releases heat.
a) What is the minimum amount of heat that can be transferred to the gas?
b) What is the maximum amount of heat that can be transferred to the gas?
(5 pont)
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P. 5693. A tetrahedron is assembled from six identical resistors each of resistance \(\displaystyle R\), and two batteries, each with a voltage of \(\displaystyle U_0\), such that one battery is connected across the vertices \(\displaystyle AB\) and the other is connected across the vertices \(\displaystyle CD\). How much heat is generated over a given time \(\displaystyle T\) across the entire network? (The internal resistance of the batteries is negligible.)
(4 pont)
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P. 5694. We use a conventional diffraction grating to produce the diffraction pattern of the light from a sodium vapour lamp. Estimate how many thin slits the grating consists of if we can just resolve the two spectral lines of the sodium lamp with wavelengths of \(\displaystyle 589.6~\mathrm{nm}\) and \(\displaystyle 590.0~\mathrm{nm}\) in the first order.
(5 pont)
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P. 5695. We illuminate a vacuum photocell with a caesium cathode using a constant-power light beam with a wavelength of 420 nm. The graph shows the intensity \(\displaystyle I\) of the photocell's photocurrent as a function of the voltage \(\displaystyle U\) between the anode and cathode.
a) What is the work function of caesium?
b) What is the minimum power of the light beam that reaches the cathode of the photocell?
(4 pont)
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P. 5696. A constant current flows through a long, straight conductor positioned vertically in vacuum. From point \(\displaystyle P\), located at a distance \(\displaystyle r_0\) from the conductor, we launch a proton onto the straight conductor in a direction perpendicular to the plane containing point \(\displaystyle P\), as shown in the figure. During its motion, the maximum distance of the proton from the conductor is \(\displaystyle 4r_0\).
a) During the motion, what is the maximum angle \(\displaystyle \vartheta\) between the direction of motion of the proton and the horizontal?
b) What is the angle \(\displaystyle \vartheta\) at the moment when the proton is at a distance \(\displaystyle 2r_0\) from the straight conductor?
(6 pont)
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